Is there a relationship between axial $U(1)$ symmetry and parity transformations? Some books, such as Maggiore, write that the axial $U(1)$ transformation is the chiral transformation. A Dirac field $\Psi \in (\frac12, \frac12)$ undergoes a chiral transformation in the following manner.
$$ \Psi \mapsto e^{i\beta\gamma_5}\Psi $$
In a chiral basis, we can write $\Psi \equiv \begin{pmatrix} \psi_L\\ \xi_R\end{pmatrix}$ where $\psi_L \in (\frac12,0)$ and $\xi_R \in (0,\frac12)$. Then,
$$ \psi_L \mapsto e^{-i\beta}\psi_L \,,\qquad \xi_R \mapsto e^{i\beta}\xi_R \,. $$
Maggiore defines chiral fields to be ones that violate the axial $U(1)$ symmetry. 
Whereas books such as Srednicki define chiral fields to be ones that violate parity. A parity transformation does the following.
$$ \begin{pmatrix} \psi_L\\ \xi_R\end{pmatrix} \mapsto \begin{pmatrix} \xi_R\\\psi_L \end{pmatrix} $$
Srednicki discusses the axial $U(1)$ symmetry in a different chapter as a global symmetry in consistent gauge theories.
So, which of the definitions of chirality is true? And if they are equivalent, how so?
 A: They are both manifestly true. Any confusion is traceable to notational overkill.
In your 2d Weyl basis where superfluous spinor indices are ignored, an axial rotation is but
$$
e^{-i\beta \sigma_3} =1\!\!1 \cos \beta -i \sigma_3 \sin \beta ,
$$
where, of course, the finite rotation $\beta=\pi/2$ amounts to the above reducing to $-i\sigma_3= -\tfrac{i}{2}(1\!\!1+i\sigma_2)\sigma_1 (1\!\!1-i\sigma_2)$, so unitarily equivalent to $-i\sigma_1$.
Likewise, 
$$
P=\sigma_1
$$
which has the same eigenvalues as the π/2 rotation above, and provides its conjugation matrix, 
$$
P e^{-i\beta \sigma_3} P= e^{i\beta \sigma_3},
$$
since $\sigma_1 \sigma_3\sigma_1=-\sigma_3$.
So, yes, axial rotations differentiate between chiral eigenstates and parity interchanges them.  You cannot have an invariant of axial rotations which is not parity invariant; and, conversely, for fermions, you won't be able to construct an parity invariant which violates the axial U(1).
We are talking about fermions here, but, parity-odd spinless states like 
pseudoscalars (which might be used to violate parity in an action) can be associated with fermion spinless bilinears which comport with this axial rotation scheme: so you might as well consider the two bases connected by $(1\!\!1-i\sigma_2)/\sqrt2 $ equivalent. 
